Generating Function for rational sequence

I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the differential equation at the end. My work so far is below. Thanks in advance!

$C_N = \frac{\beta}{N(\beta-N)}$

$N(\beta-N)C_N = \beta$

$\sum_{N=1}N(\beta-N)C_N z^N = \sum_{N=1}\beta z^N$

$\sum_{N=1}N\beta C_N z^N -\sum_{N=1}N^2 C_N z^N = \sum_{N=1}\beta z^N$

$\beta zC'(z) - (2\sum_{N=1}$ $N\choose2$ $C_N z^N$ + $\sum_{N=1}N C_Nz^N) = \beta\sum_{N=1} z^N$

$\beta zC'(z) - 2z^2C''(z) - zC'(z) = \beta\sum_{N=1} z^N$

$(\beta - 1)zC'(z) - 2z^2C''(z)=\beta \frac{1}{1-z}$

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That is an Euler equation. Your tame computer algebra system (e.g. Maxima) should finish it off. Getting the coefficients out of the resulting mess will probably just give you back the original relation you started with... –  vonbrand Feb 26 '13 at 17:33